The Adjunction Theory of Complex Projective Varieties

Preface
List of tables
Chapter 1. General background results
1.1 Some basic definitions
1.2 Surface singularities
1.3 On the singularities that arise in adjunction theory
1.4 Curves
1.5 Nefvalue results
1.6 Universal sections and discriminant varieties
1.7 Bertini theorems
1.8 Some examples
Chapter 2. Consequences of positivity
2.1 k-ampleness and k-bigness
2.2 Vanishing theorems
2.3 The Lefschetz hyperplane section theorem
2.4 The Albanese mapping in the presence of rational singularities
2.5 The Hodge index theorem and the Kodaira lemma
2.6 Rossi’s extension theorems
2.7 Theorems of Andreotti-Grauert and Griffiths
Chapter 3. The basic varieties of adjunction theory
3.1 Recognizing projective spaces and quadrics
3.2 ℙd-bundles
3.3 Special varieties arising in adjunction theory
Chapter 4. The Hilbert scheme and extremal rays
4.1 Flatness, the Hilbert scheme, and limited families
4.2 Extremal rays and the cone theorem
4.3 Varieties with nonnef canonical bundle
Chapter 5. Restrictions imposed by ample divisors
5.1 On the behavior of k-big and ample divisors under maps
5.2 Extending morphisms of ample divisors
5.3 Ample divisors with trivial pluricanonical systems
5.4 Varieties that can be ample divisors only on cones
5.5 ℙd-bundles as ample divisors
Chapter 6. Families of unbreakable rational curves
6.1 Examples
6.2 Families of unbreakable rational curves
6.3 The nonbreaking lemma
6.4 Morphisms of varieties covered by unbreakable rational curves
6.5 The classification of projective manifolds covered by lines
6.6 Some spannedness results
Chapter 7. General adjunction theory
7.1 Spectral values
7.2 Polarized pairs (ℳ, ℒ) with nefvalue > dim ℳ – l and ℳ singular
7.3 The first reduction of a singular variety
7.4 The polarization of the first reduction
7.5 The second reduction in the smooth case
7.6 Properties of the first and the second reduction
7.7 The second reduction (X, D) with KX + (n – 3) D nef
7.8 The three dimensional case
7.9 Applications
Chapter 8. Background for classical adjunction theory
8.1 Numerical implications of nonnegative Kodaira dimension
8.2 The double point formula for surfaces
8.3 Smooth double covers of irreducible quadric surfaces
8.4 Surfaces with one dimensional projection from a line
8.5 k-very ampleness
8.6 Surfaces with Castelnuovo curves as hyperplane sections
8.7 Polarized varieties (X, L) with sectional genus g(L) = h1(OX)
8.8 Spannedness of KX + (dim X)L for ample and spanned L
8.9 Polarized varieties (X, L) with sectional genus g(L) ≤ 1
8.10 Classification of varieties up to degree 4
Chapter 9. The adjunction mapping
9.1 Spannedness of adjoint bundles at singular points
9.2 The adjunction mapping
Chapter 10. Classical adjunction theory of surfaces
10.1 When the adjunction mapping has lower dimensional image
10.2 Surfaces with sectional genus g(L) ≤ 3
10.3 Very ampleness of the adjoint bundle
10.4 Very ampleness of the adjoint bundle for degree d ≥ 9
10.5 Very ampleness of the adjoint bundle when h1(OS) > 0
10.6 Very ampleness of the adjoint bundle when h1(OS) = 0
10.7 Preservation of k-very ampleness under adjunction
Chapter 11. Classical adjunction theory in dimension ≥ 3
11.1 Some results on scrolls
11.2 The adjunction mapping with a lower dimensional image
11.3 Very ampleness of the adjoint bundle
11.4 Applications to hyperelliptic curve sections
11.5 Projective normality of adjoint bundles
11.6 Manifolds of sectional genus ≤ 4
11.7 The Fano-Morin adjunction process
Chapter 12. The second reduction in dimension three
12.1 Exceptional divisors of the second reduction morphism
12.2 The structure of the second reduction
12.3 The second reduction for threefolds in ℙ5
Chapter 13. Varieties (ℳ, ℒ) with κ(Κℳ + (dim ℳ – 2)ℒ) ≥ 0
13.1 The double point formula for threefolds
13.2 The linear system |Kℳ + (n – 2)L| on the first reduction (M, L)
13.3 Some Chern inequalities for ample divisors
Chapter 14. Special varieties
14.1 Structure results for scrolls
14.2 Structure results for quadric fibrations
14.3 Varieties with small invariants
14.4 Projective manifolds with positive defect
14.5 Hyperplane sections of curves
Bibliography
Index